Deriving the Brownian Motion in the Margrabe Formula

Summary

The post addresses a step in deriving the Margrabe formula for a claim paying the positive difference between two correlated assets at maturity. After choosing the second asset as numéraire, the question asks how the ratio of the assets can be written as a geometric diffusion driven by a single standard Brownian motion, and under which probability measure that motion is defined.

The answer constructs the driver as a normalized linear combination of the two Brownian motions under the changed measure. Its normalization uses the variance of that combination, which depends on both asset volatilities and their correlation. Levy’s characterization then establishes that the normalized process is standard Brownian motion under the numéraire measure. This resolves the specific diffusion step, but the short exchange does not present the full option-pricing derivation or explore special cases such as a degenerate variance.

Key ideas

• Changing to the second asset as numéraire changes the probability measure used for the ratio dynamics.
• The ratio’s diffusion driver is a linear combination of the two Brownian motions under that measure.
• Normalizing by the combination’s volatility produces a standard Brownian motion, justified by Levy’s characterization.
• The effective variance depends on both asset volatilities and their correlation.

Tags

# Derivation of Magrabe formula


# Derivation of Magrabe formula












I'm going through the following note by Davis, link.

In chapter 3 he derives the `Magrabe` formula. I got stuck at equation $(3.16)$. We have two assets:

$$dS_i(t)=S_i(t)\sigma_idw_i(t)$$ for $i\in\{1,2\}$ and $d\langle w_1,w_2\rangle_t = \rho dt$. The payoff we are interested in is the following:

$$C(0,s_1,s_2)=E[\max{(S_1(T)-S_2(T),0)}]$$

The idea is to perform a change of measure, using $S_2(T)$ as numéraire. We end with a new measure $\tilde{P}$ with $$d\tilde{w}_2=dw_2-\sigma_2dt$$ and $$d\tilde{w}_1=dw_1-\rho\sigma_2dt$$ both brownian motion under $\tilde{P}$. Defining $Y:=\frac{S_1}{S_2}$ one can show that:

$$dY=Y(\sigma_1d\tilde{w}_1-\sigma_2d\tilde{w}_2)$$

The authors claims that this can be writen as:

$$dY=Y\sigma dw\tag{3.16}$$

where $w$ is a standard brownian motion and $\sigma = \sqrt{\sigma^2_1+\sigma^2_1-2\sigma_1\sigma_2\rho}$.

How do we get $(3.16)$ and the brownian motion $w$, especially under which measure?

## Answer by Gordon (score 2, accepted)

https://quant.stackexchange.com/a/21218

Let \begin{align*} w_t = \frac{1}{\sqrt{\sigma_1^2+\sigma_2^2 -2\sigma_1\sigma_2 \rho}}(\sigma_1\tilde{w}_t^1-\sigma_2\tilde{w}_t^2). \end{align*} Then, using Levy's characterization, we can show that $\{w_t \mid t \geq 0\}$ is a standard Brownian motion.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.